LMFDB - Embedded newform 799.2.b.a.424.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [799,2,Mod(424,799)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(799, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("799.424");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 799 = 17 \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	799.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$6.38004712150$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			424.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	799.424
Dual form			799.2.b.a.424.2

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.00000i q^{3} -1.00000 q^{4} +2.00000i q^{5} -2.00000i q^{6} +2.00000i q^{7} -3.00000 q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.00000i q^{3} -1.00000 q^{4} +2.00000i q^{5} -2.00000i q^{6} +2.00000i q^{7} -3.00000 q^{8} -1.00000 q^{9} +2.00000i q^{10} +2.00000i q^{12} -6.00000 q^{13} +2.00000i q^{14} +4.00000 q^{15} -1.00000 q^{16} +(1.00000 + 4.00000i) q^{17} -1.00000 q^{18} -8.00000 q^{19} -2.00000i q^{20} +4.00000 q^{21} +8.00000i q^{23} +6.00000i q^{24} +1.00000 q^{25} -6.00000 q^{26} -4.00000i q^{27} -2.00000i q^{28} +6.00000i q^{29} +4.00000 q^{30} -4.00000i q^{31} +5.00000 q^{32} +(1.00000 + 4.00000i) q^{34} -4.00000 q^{35} +1.00000 q^{36} -8.00000 q^{38} +12.0000i q^{39} -6.00000i q^{40} -2.00000i q^{41} +4.00000 q^{42} -2.00000i q^{45} +8.00000i q^{46} -1.00000 q^{47} +2.00000i q^{48} +3.00000 q^{49} +1.00000 q^{50} +(8.00000 - 2.00000i) q^{51} +6.00000 q^{52} +6.00000 q^{53} -4.00000i q^{54} -6.00000i q^{56} +16.0000i q^{57} +6.00000i q^{58} -12.0000 q^{59} -4.00000 q^{60} -8.00000i q^{61} -4.00000i q^{62} -2.00000i q^{63} +7.00000 q^{64} -12.0000i q^{65} -8.00000 q^{67} +(-1.00000 - 4.00000i) q^{68} +16.0000 q^{69} -4.00000 q^{70} +10.0000i q^{71} +3.00000 q^{72} -2.00000i q^{73} -2.00000i q^{75} +8.00000 q^{76} +12.0000i q^{78} +14.0000i q^{79} -2.00000i q^{80} -11.0000 q^{81} -2.00000i q^{82} -4.00000 q^{83} -4.00000 q^{84} +(-8.00000 + 2.00000i) q^{85} +12.0000 q^{87} +10.0000 q^{89} -2.00000i q^{90} -12.0000i q^{91} -8.00000i q^{92} -8.00000 q^{93} -1.00000 q^{94} -16.0000i q^{95} -10.0000i q^{96} +8.00000i q^{97} +3.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 2 q^{4} - 6 q^{8} - 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 - 2 * q^4 - 6 * q^8 - 2 * q^9	$ 2 q + 2 q^{2} - 2 q^{4} - 6 q^{8} - 2 q^{9} - 12 q^{13} + 8 q^{15} - 2 q^{16} + 2 q^{17} - 2 q^{18} - 16 q^{19} + 8 q^{21} + 2 q^{25} - 12 q^{26} + 8 q^{30} + 10 q^{32} + 2 q^{34} - 8 q^{35} + 2 q^{36} - 16 q^{38} + 8 q^{42} - 2 q^{47} + 6 q^{49} + 2 q^{50} + 16 q^{51} + 12 q^{52} + 12 q^{53} - 24 q^{59} - 8 q^{60} + 14 q^{64} - 16 q^{67} - 2 q^{68} + 32 q^{69} - 8 q^{70} + 6 q^{72} + 16 q^{76} - 22 q^{81} - 8 q^{83} - 8 q^{84} - 16 q^{85} + 24 q^{87} + 20 q^{89} - 16 q^{93} - 2 q^{94} + 6 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 - 2 * q^4 - 6 * q^8 - 2 * q^9 - 12 * q^13 + 8 * q^15 - 2 * q^16 + 2 * q^17 - 2 * q^18 - 16 * q^19 + 8 * q^21 + 2 * q^25 - 12 * q^26 + 8 * q^30 + 10 * q^32 + 2 * q^34 - 8 * q^35 + 2 * q^36 - 16 * q^38 + 8 * q^42 - 2 * q^47 + 6 * q^49 + 2 * q^50 + 16 * q^51 + 12 * q^52 + 12 * q^53 - 24 * q^59 - 8 * q^60 + 14 * q^64 - 16 * q^67 - 2 * q^68 + 32 * q^69 - 8 * q^70 + 6 * q^72 + 16 * q^76 - 22 * q^81 - 8 * q^83 - 8 * q^84 - 16 * q^85 + 24 * q^87 + 20 * q^89 - 16 * q^93 - 2 * q^94 + 6 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/799\mathbb{Z}\right)^\times$.

$n$	$52$	$377$
$\chi(n)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	1.00000			0.707107			0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000			0.707107			0.353553	+	0.935414i	$0.384973\pi$
$3$		−	2.00000i		−	1.15470i	−0.816497	−	0.577350i	$-0.804087\pi$
$3$		−	2.00000i		−	1.15470i	0.816497	−	0.577350i	$-0.195913\pi$
$4$	−1.00000			−0.500000
$5$			2.00000i			0.894427i	0.894427	+	0.447214i	$0.147584\pi$
$5$			2.00000i			0.894427i	−0.894427	+	0.447214i	$0.852416\pi$
$6$		−	2.00000i		−	0.816497i
$7$			2.00000i			0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$			2.00000i			0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	−3.00000			−1.06066
$9$	−1.00000			−0.333333
$10$			2.00000i			0.632456i
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$			2.00000i			0.577350i
$13$	−6.00000			−1.66410			−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000			−1.66410			−0.832050	+	0.554700i	$0.812833\pi$
$14$			2.00000i			0.534522i
$15$	4.00000			1.03280
$16$	−1.00000			−0.250000
$17$	1.00000	+	4.00000i	0.242536	+	0.970143i
$17$	1.00000	+	4.00000i	0.242536	+	0.970143i
$18$	−1.00000			−0.235702
$19$	−8.00000			−1.83533			−0.917663	−	0.397360i	$-0.869927\pi$
$19$	−8.00000			−1.83533			−0.917663	+	0.397360i	$0.869927\pi$
$20$		−	2.00000i		−	0.447214i
$21$	4.00000			0.872872
$22$		0			0
$23$			8.00000i			1.66812i	0.551677	+	0.834058i	$0.313988\pi$
$23$			8.00000i			1.66812i	−0.551677	+	0.834058i	$0.686012\pi$
$24$			6.00000i			1.22474i
$25$	1.00000			0.200000
$26$	−6.00000			−1.17670
$27$		−	4.00000i		−	0.769800i
$28$		−	2.00000i		−	0.377964i
$29$			6.00000i			1.11417i	0.830455	+	0.557086i	$0.188081\pi$
$29$			6.00000i			1.11417i	−0.830455	+	0.557086i	$0.811919\pi$
$30$	4.00000			0.730297
$31$		−	4.00000i		−	0.718421i	−0.933257	−	0.359211i	$-0.883046\pi$
$31$		−	4.00000i		−	0.718421i	0.933257	−	0.359211i	$-0.116954\pi$
$32$	5.00000			0.883883
$33$		0			0
$34$	1.00000	+	4.00000i	0.171499	+	0.685994i
$35$	−4.00000			−0.676123
$36$	1.00000			0.166667
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$	−8.00000			−1.29777
$39$			12.0000i			1.92154i
$40$		−	6.00000i		−	0.948683i
$41$		−	2.00000i		−	0.312348i	−0.987730	−	0.156174i	$-0.950084\pi$
$41$		−	2.00000i		−	0.312348i	0.987730	−	0.156174i	$-0.0499160\pi$
$42$	4.00000			0.617213
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$		0			0
$45$		−	2.00000i		−	0.298142i
$46$			8.00000i			1.17954i
$47$	−1.00000			−0.145865
$47$	−1.00000			−0.145865
$48$			2.00000i			0.288675i
$49$	3.00000			0.428571
$50$	1.00000			0.141421
$51$	8.00000	−	2.00000i	1.12022	−	0.280056i
$52$	6.00000			0.832050
$53$	6.00000			0.824163			0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000			0.824163			0.412082	+	0.911147i	$0.364802\pi$
$54$		−	4.00000i		−	0.544331i
$55$		0			0
$56$		−	6.00000i		−	0.801784i
$57$			16.0000i			2.11925i
$58$			6.00000i			0.787839i
$59$	−12.0000			−1.56227			−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000			−1.56227			−0.781133	+	0.624364i	$0.785358\pi$
$60$	−4.00000			−0.516398
$61$		−	8.00000i		−	1.02430i	−0.858898	−	0.512148i	$-0.828850\pi$
$61$		−	8.00000i		−	1.02430i	0.858898	−	0.512148i	$-0.171150\pi$
$62$		−	4.00000i		−	0.508001i
$63$		−	2.00000i		−	0.251976i
$64$	7.00000			0.875000
$65$		−	12.0000i		−	1.48842i
$66$		0			0
$67$	−8.00000			−0.977356			−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000			−0.977356			−0.488678	+	0.872464i	$0.662521\pi$
$68$	−1.00000	−	4.00000i	−0.121268	−	0.485071i
$69$	16.0000			1.92617
$70$	−4.00000			−0.478091
$71$			10.0000i			1.18678i	0.804914	+	0.593391i	$0.202211\pi$
$71$			10.0000i			1.18678i	−0.804914	+	0.593391i	$0.797789\pi$
$72$	3.00000			0.353553
$73$		−	2.00000i		−	0.234082i	−0.993127	−	0.117041i	$-0.962659\pi$
$73$		−	2.00000i		−	0.234082i	0.993127	−	0.117041i	$-0.0373409\pi$
$74$		0			0
$75$		−	2.00000i		−	0.230940i
$76$	8.00000			0.917663
$77$		0			0
$78$			12.0000i			1.35873i
$79$			14.0000i			1.57512i	0.616236	+	0.787562i	$0.288657\pi$
$79$			14.0000i			1.57512i	−0.616236	+	0.787562i	$0.711343\pi$
$80$		−	2.00000i		−	0.223607i
$81$	−11.0000			−1.22222
$82$		−	2.00000i		−	0.220863i
$83$	−4.00000			−0.439057			−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000			−0.439057			−0.219529	+	0.975606i	$0.570452\pi$
$84$	−4.00000			−0.436436
$85$	−8.00000	+	2.00000i	−0.867722	+	0.216930i
$86$		0			0
$87$	12.0000			1.28654
$88$		0			0
$89$	10.0000			1.06000			0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000			1.06000			0.529999	+	0.847998i	$0.322192\pi$
$90$		−	2.00000i		−	0.210819i
$91$		−	12.0000i		−	1.25794i
$92$		−	8.00000i		−	0.834058i
$93$	−8.00000			−0.829561
$94$	−1.00000			−0.103142
$95$		−	16.0000i		−	1.64157i
$96$		−	10.0000i		−	1.02062i
$97$			8.00000i			0.812277i	0.913812	+	0.406138i	$0.133125\pi$
$97$			8.00000i			0.812277i	−0.913812	+	0.406138i	$0.866875\pi$
$98$	3.00000			0.303046
$99$		0			0
$100$	−1.00000			−0.100000
$101$	−6.00000			−0.597022			−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000			−0.597022			−0.298511	+	0.954406i	$0.596490\pi$
$102$	8.00000	−	2.00000i	0.792118	−	0.198030i
$103$	8.00000			0.788263			0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000			0.788263			0.394132	+	0.919054i	$0.371045\pi$
$104$	18.0000			1.76505
$105$			8.00000i			0.780720i
$106$	6.00000			0.582772
$107$		−	4.00000i		−	0.386695i	−0.981130	−	0.193347i	$-0.938066\pi$
$107$		−	4.00000i		−	0.386695i	0.981130	−	0.193347i	$-0.0619344\pi$
$108$			4.00000i			0.384900i
$109$		−	10.0000i		−	0.957826i	−0.877862	−	0.478913i	$-0.841031\pi$
$109$		−	10.0000i		−	0.957826i	0.877862	−	0.478913i	$-0.158969\pi$
$110$		0			0
$111$		0			0
$112$		−	2.00000i		−	0.188982i
$113$		−	14.0000i		−	1.31701i	−0.752577	−	0.658505i	$-0.771189\pi$
$113$		−	14.0000i		−	1.31701i	0.752577	−	0.658505i	$-0.228811\pi$
$114$			16.0000i			1.49854i
$115$	−16.0000			−1.49201
$116$		−	6.00000i		−	0.557086i
$117$	6.00000			0.554700
$118$	−12.0000			−1.10469
$119$	−8.00000	+	2.00000i	−0.733359	+	0.183340i
$120$	−12.0000			−1.09545
$121$	11.0000			1.00000
$122$		−	8.00000i		−	0.724286i
$123$	−4.00000			−0.360668
$124$			4.00000i			0.359211i
$125$			12.0000i			1.07331i
$126$		−	2.00000i		−	0.178174i
$127$	−12.0000			−1.06483			−0.532414	−	0.846484i	$-0.678715\pi$
$127$	−12.0000			−1.06483			−0.532414	+	0.846484i	$0.678715\pi$
$128$	−3.00000			−0.265165
$129$		0			0
$130$		−	12.0000i		−	1.05247i
$131$		−	6.00000i		−	0.524222i	−0.965038	−	0.262111i	$-0.915581\pi$
$131$		−	6.00000i		−	0.524222i	0.965038	−	0.262111i	$-0.0844187\pi$
$132$		0			0
$133$		−	16.0000i		−	1.38738i
$134$	−8.00000			−0.691095
$135$	8.00000			0.688530
$136$	−3.00000	−	12.0000i	−0.257248	−	1.02899i
$137$	−18.0000			−1.53784			−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000			−1.53784			−0.768922	+	0.639343i	$0.779207\pi$
$138$	16.0000			1.36201
$139$		−	8.00000i		−	0.678551i	−0.940687	−	0.339276i	$-0.889818\pi$
$139$		−	8.00000i		−	0.678551i	0.940687	−	0.339276i	$-0.110182\pi$
$140$	4.00000			0.338062
$141$			2.00000i			0.168430i
$142$			10.0000i			0.839181i
$143$		0			0
$144$	1.00000			0.0833333
$145$	−12.0000			−0.996546
$146$		−	2.00000i		−	0.165521i
$147$		−	6.00000i		−	0.494872i
$148$		0			0
$149$	18.0000			1.47462			0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000			1.47462			0.737309	+	0.675556i	$0.236096\pi$
$150$		−	2.00000i		−	0.163299i
$151$	−16.0000			−1.30206			−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000			−1.30206			−0.651031	+	0.759051i	$0.725663\pi$
$152$	24.0000			1.94666
$153$	−1.00000	−	4.00000i	−0.0808452	−	0.323381i
$154$		0			0
$155$	8.00000			0.642575
$156$		−	12.0000i		−	0.960769i
$157$	−14.0000			−1.11732			−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000			−1.11732			−0.558661	+	0.829396i	$0.688685\pi$
$158$			14.0000i			1.11378i
$159$		−	12.0000i		−	0.951662i
$160$			10.0000i			0.790569i
$161$	−16.0000			−1.26098
$162$	−11.0000			−0.864242
$163$			8.00000i			0.626608i	0.949653	+	0.313304i	$0.101436\pi$
$163$			8.00000i			0.626608i	−0.949653	+	0.313304i	$0.898564\pi$
$164$			2.00000i			0.156174i
$165$		0			0
$166$	−4.00000			−0.310460
$167$		−	12.0000i		−	0.928588i	−0.885681	−	0.464294i	$-0.846308\pi$
$167$		−	12.0000i		−	0.928588i	0.885681	−	0.464294i	$-0.153692\pi$
$168$	−12.0000			−0.925820
$169$	23.0000			1.76923
$170$	−8.00000	+	2.00000i	−0.613572	+	0.153393i
$171$	8.00000			0.611775
$172$		0			0
$173$			8.00000i			0.608229i	0.952636	+	0.304114i	$0.0983605\pi$
$173$			8.00000i			0.608229i	−0.952636	+	0.304114i	$0.901639\pi$
$174$	12.0000			0.909718
$175$			2.00000i			0.151186i
$176$		0			0
$177$			24.0000i			1.80395i
$178$	10.0000			0.749532
$179$	12.0000			0.896922			0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000			0.896922			0.448461	+	0.893802i	$0.351972\pi$
$180$			2.00000i			0.149071i
$181$			2.00000i			0.148659i	0.997234	+	0.0743294i	$0.0236816\pi$
$181$			2.00000i			0.148659i	−0.997234	+	0.0743294i	$0.976318\pi$
$182$		−	12.0000i		−	0.889499i
$183$	−16.0000			−1.18275
$184$		−	24.0000i		−	1.76930i
$185$		0			0
$186$	−8.00000			−0.586588
$187$		0			0
$188$	1.00000			0.0729325
$189$	8.00000			0.581914
$190$		−	16.0000i		−	1.16076i
$191$	8.00000			0.578860			0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000			0.578860			0.289430	+	0.957199i	$0.406534\pi$
$192$		−	14.0000i		−	1.01036i
$193$			2.00000i			0.143963i	0.997406	+	0.0719816i	$0.0229323\pi$
$193$			2.00000i			0.143963i	−0.997406	+	0.0719816i	$0.977068\pi$
$194$			8.00000i			0.574367i
$195$	−24.0000			−1.71868
$196$	−3.00000			−0.214286
$197$			12.0000i			0.854965i	0.904024	+	0.427482i	$0.140599\pi$
$197$			12.0000i			0.854965i	−0.904024	+	0.427482i	$0.859401\pi$
$198$		0			0
$199$			16.0000i			1.13421i	0.823646	+	0.567105i	$0.191937\pi$
$199$			16.0000i			1.13421i	−0.823646	+	0.567105i	$0.808063\pi$
$200$	−3.00000			−0.212132
$201$			16.0000i			1.12855i
$202$	−6.00000			−0.422159
$203$	−12.0000			−0.842235
$204$	−8.00000	+	2.00000i	−0.560112	+	0.140028i
$205$	4.00000			0.279372
$206$	8.00000			0.557386
$207$		−	8.00000i		−	0.556038i
$208$	6.00000			0.416025
$209$		0			0
$210$			8.00000i			0.552052i
$211$		−	4.00000i		−	0.275371i	−0.990476	−	0.137686i	$-0.956034\pi$
$211$		−	4.00000i		−	0.275371i	0.990476	−	0.137686i	$-0.0439664\pi$
$212$	−6.00000			−0.412082
$213$	20.0000			1.37038
$214$		−	4.00000i		−	0.273434i
$215$		0			0
$216$			12.0000i			0.816497i
$217$	8.00000			0.543075
$218$		−	10.0000i		−	0.677285i
$219$	−4.00000			−0.270295
$220$		0			0
$221$	−6.00000	−	24.0000i	−0.403604	−	1.61441i
$222$		0			0
$223$	8.00000			0.535720			0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000			0.535720			0.267860	+	0.963458i	$0.413684\pi$
$224$			10.0000i			0.668153i
$225$	−1.00000			−0.0666667
$226$		−	14.0000i		−	0.931266i
$227$			12.0000i			0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$			12.0000i			0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$		−	16.0000i		−	1.05963i
$229$	10.0000			0.660819			0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000			0.660819			0.330409	+	0.943838i	$0.392813\pi$
$230$	−16.0000			−1.05501
$231$		0			0
$232$		−	18.0000i		−	1.18176i
$233$			18.0000i			1.17922i	0.807688	+	0.589610i	$0.200718\pi$
$233$			18.0000i			1.17922i	−0.807688	+	0.589610i	$0.799282\pi$
$234$	6.00000			0.392232
$235$		−	2.00000i		−	0.130466i
$236$	12.0000			0.781133
$237$	28.0000			1.81880
$238$	−8.00000	+	2.00000i	−0.518563	+	0.129641i
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$	−4.00000			−0.258199
$241$			12.0000i			0.772988i	0.922292	+	0.386494i	$0.126314\pi$
$241$			12.0000i			0.772988i	−0.922292	+	0.386494i	$0.873686\pi$
$242$	11.0000			0.707107
$243$			10.0000i			0.641500i
$244$			8.00000i			0.512148i
$245$			6.00000i			0.383326i
$246$	−4.00000			−0.255031
$247$	48.0000			3.05417
$248$			12.0000i			0.762001i
$249$			8.00000i			0.506979i
$250$			12.0000i			0.758947i
$251$	4.00000			0.252478			0.126239	−	0.992000i	$-0.459709\pi$
$251$	4.00000			0.252478			0.126239	+	0.992000i	$0.459709\pi$
$252$			2.00000i			0.125988i
$253$		0			0
$254$	−12.0000			−0.752947
$255$	4.00000	+	16.0000i	0.250490	+	1.00196i
$256$	−17.0000			−1.06250
$257$	22.0000			1.37232			0.686161	−	0.727450i	$-0.259294\pi$
$257$	22.0000			1.37232			0.686161	+	0.727450i	$0.259294\pi$
$258$		0			0
$259$		0			0
$260$			12.0000i			0.744208i
$261$		−	6.00000i		−	0.371391i
$262$		−	6.00000i		−	0.370681i
$263$	8.00000			0.493301			0.246651	−	0.969104i	$-0.420670\pi$
$263$	8.00000			0.493301			0.246651	+	0.969104i	$0.420670\pi$
$264$		0			0
$265$			12.0000i			0.737154i
$266$		−	16.0000i		−	0.981023i
$267$		−	20.0000i		−	1.22398i
$268$	8.00000			0.488678
$269$			32.0000i			1.95107i	0.219834	+	0.975537i	$0.429448\pi$
$269$			32.0000i			1.95107i	−0.219834	+	0.975537i	$0.570552\pi$
$270$	8.00000			0.486864
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$	−1.00000	−	4.00000i	−0.0606339	−	0.242536i
$273$	−24.0000			−1.45255
$274$	−18.0000			−1.08742
$275$		0			0
$276$	−16.0000			−0.963087
$277$			28.0000i			1.68236i	0.540758	+	0.841178i	$0.318138\pi$
$277$			28.0000i			1.68236i	−0.540758	+	0.841178i	$0.681862\pi$
$278$		−	8.00000i		−	0.479808i
$279$			4.00000i			0.239474i
$280$	12.0000			0.717137
$281$	10.0000			0.596550			0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000			0.596550			0.298275	+	0.954480i	$0.403589\pi$
$282$			2.00000i			0.119098i
$283$		−	26.0000i		−	1.54554i	−0.634686	−	0.772770i	$-0.718871\pi$
$283$		−	26.0000i		−	1.54554i	0.634686	−	0.772770i	$-0.281129\pi$
$284$		−	10.0000i		−	0.593391i
$285$	−32.0000			−1.89552
$286$		0			0
$287$	4.00000			0.236113
$288$	−5.00000			−0.294628
$289$	−15.0000	+	8.00000i	−0.882353	+	0.470588i
$290$	−12.0000			−0.704664
$291$	16.0000			0.937937
$292$			2.00000i			0.117041i
$293$	30.0000			1.75262			0.876309	−	0.481749i	$-0.159998\pi$
$293$	30.0000			1.75262			0.876309	+	0.481749i	$0.159998\pi$
$294$		−	6.00000i		−	0.349927i
$295$		−	24.0000i		−	1.39733i
$296$		0			0
$297$		0			0
$298$	18.0000			1.04271
$299$		−	48.0000i		−	2.77591i
$300$			2.00000i			0.115470i
$301$		0			0
$302$	−16.0000			−0.920697
$303$			12.0000i			0.689382i
$304$	8.00000			0.458831
$305$	16.0000			0.916157
$306$	−1.00000	−	4.00000i	−0.0571662	−	0.228665i
$307$	28.0000			1.59804			0.799022	−	0.601302i	$-0.205351\pi$
$307$	28.0000			1.59804			0.799022	+	0.601302i	$0.205351\pi$
$308$		0			0
$309$		−	16.0000i		−	0.910208i
$310$	8.00000			0.454369
$311$		−	16.0000i		−	0.907277i	−0.891186	−	0.453638i	$-0.850126\pi$
$311$		−	16.0000i		−	0.907277i	0.891186	−	0.453638i	$-0.149874\pi$
$312$		−	36.0000i		−	2.03810i
$313$			14.0000i			0.791327i	0.918396	+	0.395663i	$0.129485\pi$
$313$			14.0000i			0.791327i	−0.918396	+	0.395663i	$0.870515\pi$
$314$	−14.0000			−0.790066
$315$	4.00000			0.225374
$316$		−	14.0000i		−	0.787562i
$317$		−	6.00000i		−	0.336994i	−0.985702	−	0.168497i	$-0.946109\pi$
$317$		−	6.00000i		−	0.336994i	0.985702	−	0.168497i	$-0.0538913\pi$
$318$		−	12.0000i		−	0.672927i
$319$		0			0
$320$			14.0000i			0.782624i
$321$	−8.00000			−0.446516
$322$	−16.0000			−0.891645
$323$	−8.00000	−	32.0000i	−0.445132	−	1.78053i
$324$	11.0000			0.611111
$325$	−6.00000			−0.332820
$326$			8.00000i			0.443079i
$327$	−20.0000			−1.10600
$328$			6.00000i			0.331295i
$329$		−	2.00000i		−	0.110264i
$330$		0			0
$331$	4.00000			0.219860			0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000			0.219860			0.109930	+	0.993939i	$0.464937\pi$
$332$	4.00000			0.219529
$333$		0			0
$334$		−	12.0000i		−	0.656611i
$335$		−	16.0000i		−	0.874173i
$336$	−4.00000			−0.218218
$337$			36.0000i			1.96104i	0.196407	+	0.980522i	$0.437073\pi$
$337$			36.0000i			1.96104i	−0.196407	+	0.980522i	$0.562927\pi$
$338$	23.0000			1.25104
$339$	−28.0000			−1.52075
$340$	8.00000	−	2.00000i	0.433861	−	0.108465i
$341$		0			0
$342$	8.00000			0.432590
$343$			20.0000i			1.07990i
$344$		0			0
$345$			32.0000i			1.72282i
$346$			8.00000i			0.430083i
$347$			22.0000i			1.18102i	0.807030	+	0.590511i	$0.201074\pi$
$347$			22.0000i			1.18102i	−0.807030	+	0.590511i	$0.798926\pi$
$348$	−12.0000			−0.643268
$349$	−18.0000			−0.963518			−0.481759	−	0.876304i	$-0.660002\pi$
$349$	−18.0000			−0.963518			−0.481759	+	0.876304i	$0.660002\pi$
$350$			2.00000i			0.106904i
$351$			24.0000i			1.28103i
$352$		0			0
$353$	−18.0000			−0.958043			−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000			−0.958043			−0.479022	+	0.877803i	$0.659008\pi$
$354$			24.0000i			1.27559i
$355$	−20.0000			−1.06149
$356$	−10.0000			−0.529999
$357$	4.00000	+	16.0000i	0.211702	+	0.846810i
$358$	12.0000			0.634220
$359$	−24.0000			−1.26667			−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000			−1.26667			−0.633336	+	0.773877i	$0.718315\pi$
$360$			6.00000i			0.316228i
$361$	45.0000			2.36842
$362$			2.00000i			0.105118i
$363$		−	22.0000i		−	1.15470i
$364$			12.0000i			0.628971i
$365$	4.00000			0.209370
$366$	−16.0000			−0.836333
$367$		−	12.0000i		−	0.626395i	−0.949688	−	0.313197i	$-0.898600\pi$
$367$		−	12.0000i		−	0.626395i	0.949688	−	0.313197i	$-0.101400\pi$
$368$		−	8.00000i		−	0.417029i
$369$			2.00000i			0.104116i
$370$		0			0
$371$			12.0000i			0.623009i
$372$	8.00000			0.414781
$373$	−14.0000			−0.724893			−0.362446	−	0.932005i	$-0.618058\pi$
$373$	−14.0000			−0.724893			−0.362446	+	0.932005i	$0.618058\pi$
$374$		0			0
$375$	24.0000			1.23935
$376$	3.00000			0.154713
$377$		−	36.0000i		−	1.85409i
$378$	8.00000			0.411476
$379$			10.0000i			0.513665i	0.966456	+	0.256833i	$0.0826790\pi$
$379$			10.0000i			0.513665i	−0.966456	+	0.256833i	$0.917321\pi$
$380$			16.0000i			0.820783i
$381$			24.0000i			1.22956i
$382$	8.00000			0.409316
$383$	16.0000			0.817562			0.408781	−	0.912633i	$-0.365954\pi$
$383$	16.0000			0.817562			0.408781	+	0.912633i	$0.365954\pi$
$384$			6.00000i			0.306186i
$385$		0			0
$386$			2.00000i			0.101797i
$387$		0			0
$388$		−	8.00000i		−	0.406138i
$389$	18.0000			0.912636			0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000			0.912636			0.456318	+	0.889817i	$0.349168\pi$
$390$	−24.0000			−1.21529
$391$	−32.0000	+	8.00000i	−1.61831	+	0.404577i
$392$	−9.00000			−0.454569
$393$	−12.0000			−0.605320
$394$			12.0000i			0.604551i
$395$	−28.0000			−1.40883
$396$		0			0
$397$			32.0000i			1.60603i	0.595956	+	0.803017i	$0.296773\pi$
$397$			32.0000i			1.60603i	−0.595956	+	0.803017i	$0.703227\pi$
$398$			16.0000i			0.802008i
$399$	−32.0000			−1.60200
$400$	−1.00000			−0.0500000
$401$		−	28.0000i		−	1.39825i	−0.714998	−	0.699127i	$-0.753572\pi$
$401$		−	28.0000i		−	1.39825i	0.714998	−	0.699127i	$-0.246428\pi$
$402$			16.0000i			0.798007i
$403$			24.0000i			1.19553i
$404$	6.00000			0.298511
$405$		−	22.0000i		−	1.09319i
$406$	−12.0000			−0.595550
$407$		0			0
$408$	−24.0000	+	6.00000i	−1.18818	+	0.297044i
$409$	−38.0000			−1.87898			−0.939490	−	0.342578i	$-0.888700\pi$
$409$	−38.0000			−1.87898			−0.939490	+	0.342578i	$0.888700\pi$
$410$	4.00000			0.197546
$411$			36.0000i			1.77575i
$412$	−8.00000			−0.394132
$413$		−	24.0000i		−	1.18096i
$414$		−	8.00000i		−	0.393179i
$415$		−	8.00000i		−	0.392705i
$416$	−30.0000			−1.47087
$417$	−16.0000			−0.783523
$418$		0			0
$419$		−	12.0000i		−	0.586238i	−0.956076	−	0.293119i	$-0.905307\pi$
$419$		−	12.0000i		−	0.586238i	0.956076	−	0.293119i	$-0.0946933\pi$
$420$		−	8.00000i		−	0.390360i
$421$	−6.00000			−0.292422			−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000			−0.292422			−0.146211	+	0.989253i	$0.546708\pi$
$422$		−	4.00000i		−	0.194717i
$423$	1.00000			0.0486217
$424$	−18.0000			−0.874157
$425$	1.00000	+	4.00000i	0.0485071	+	0.194029i
$426$	20.0000			0.969003
$427$	16.0000			0.774294
$428$			4.00000i			0.193347i
$429$		0			0
$430$		0			0
$431$		−	18.0000i		−	0.867029i	−0.901146	−	0.433515i	$-0.857273\pi$
$431$		−	18.0000i		−	0.867029i	0.901146	−	0.433515i	$-0.142727\pi$
$432$			4.00000i			0.192450i
$433$	14.0000			0.672797			0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000			0.672797			0.336399	+	0.941720i	$0.390791\pi$
$434$	8.00000			0.384012
$435$			24.0000i			1.15071i
$436$			10.0000i			0.478913i
$437$		−	64.0000i		−	3.06154i
$438$	−4.00000			−0.191127
$439$			6.00000i			0.286364i	0.989696	+	0.143182i	$0.0457335\pi$
$439$			6.00000i			0.286364i	−0.989696	+	0.143182i	$0.954267\pi$
$440$		0			0
$441$	−3.00000			−0.142857
$442$	−6.00000	−	24.0000i	−0.285391	−	1.14156i
$443$	−24.0000			−1.14027			−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000			−1.14027			−0.570137	+	0.821549i	$0.693110\pi$
$444$		0			0
$445$			20.0000i			0.948091i
$446$	8.00000			0.378811
$447$		−	36.0000i		−	1.70274i
$448$			14.0000i			0.661438i
$449$		−	18.0000i		−	0.849473i	−0.905317	−	0.424736i	$-0.860367\pi$
$449$		−	18.0000i		−	0.849473i	0.905317	−	0.424736i	$-0.139633\pi$
$450$	−1.00000			−0.0471405
$451$		0			0
$452$			14.0000i			0.658505i
$453$			32.0000i			1.50349i
$454$			12.0000i			0.563188i
$455$	24.0000			1.12514
$456$		−	48.0000i		−	2.24781i
$457$	10.0000			0.467780			0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000			0.467780			0.233890	+	0.972263i	$0.424854\pi$
$458$	10.0000			0.467269
$459$	16.0000	−	4.00000i	0.746816	−	0.186704i
$460$	16.0000			0.746004
$461$	14.0000			0.652045			0.326023	−	0.945362i	$-0.394291\pi$
$461$	14.0000			0.652045			0.326023	+	0.945362i	$0.394291\pi$
$462$		0			0
$463$	32.0000			1.48717			0.743583	−	0.668644i	$-0.233125\pi$
$463$	32.0000			1.48717			0.743583	+	0.668644i	$0.233125\pi$
$464$		−	6.00000i		−	0.278543i
$465$		−	16.0000i		−	0.741982i
$466$			18.0000i			0.833834i
$467$	4.00000			0.185098			0.0925490	−	0.995708i	$-0.470499\pi$
$467$	4.00000			0.185098			0.0925490	+	0.995708i	$0.470499\pi$
$468$	−6.00000			−0.277350
$469$		−	16.0000i		−	0.738811i
$470$		−	2.00000i		−	0.0922531i
$471$			28.0000i			1.29017i
$472$	36.0000			1.65703
$473$		0			0
$474$	28.0000			1.28608
$475$	−8.00000			−0.367065
$476$	8.00000	−	2.00000i	0.366679	−	0.0916698i
$477$	−6.00000			−0.274721
$478$		0			0
$479$			6.00000i			0.274147i	0.990561	+	0.137073i	$0.0437697\pi$
$479$			6.00000i			0.274147i	−0.990561	+	0.137073i	$0.956230\pi$
$480$	20.0000			0.912871
$481$		0			0
$482$			12.0000i			0.546585i
$483$			32.0000i			1.45605i
$484$	−11.0000			−0.500000
$485$	−16.0000			−0.726523
$486$			10.0000i			0.453609i
$487$		−	26.0000i		−	1.17817i	−0.808070	−	0.589086i	$-0.799488\pi$
$487$		−	26.0000i		−	1.17817i	0.808070	−	0.589086i	$-0.200512\pi$
$488$			24.0000i			1.08643i
$489$	16.0000			0.723545
$490$			6.00000i			0.271052i
$491$	−36.0000			−1.62466			−0.812329	−	0.583200i	$-0.801800\pi$
$491$	−36.0000			−1.62466			−0.812329	+	0.583200i	$0.801800\pi$
$492$	4.00000			0.180334
$493$	−24.0000	+	6.00000i	−1.08091	+	0.270226i
$494$	48.0000			2.15962
$495$		0			0
$496$			4.00000i			0.179605i
$497$	−20.0000			−0.897123
$498$			8.00000i			0.358489i
$499$			16.0000i			0.716258i	0.933672	+	0.358129i	$0.116585\pi$
$499$			16.0000i			0.716258i	−0.933672	+	0.358129i	$0.883415\pi$
$500$		−	12.0000i		−	0.536656i
$501$	−24.0000			−1.07224
$502$	4.00000			0.178529
$503$		−	12.0000i		−	0.535054i	−0.963550	−	0.267527i	$-0.913794\pi$
$503$		−	12.0000i		−	0.535054i	0.963550	−	0.267527i	$-0.0862064\pi$
$504$			6.00000i			0.267261i
$505$		−	12.0000i		−	0.533993i
$506$		0			0
$507$		−	46.0000i		−	2.04293i
$508$	12.0000			0.532414
$509$	−18.0000			−0.797836			−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000			−0.797836			−0.398918	+	0.916987i	$0.630614\pi$
$510$	4.00000	+	16.0000i	0.177123	+	0.708492i
$511$	4.00000			0.176950
$512$	−11.0000			−0.486136
$513$			32.0000i			1.41283i
$514$	22.0000			0.970378
$515$			16.0000i			0.705044i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	16.0000			0.702322
$520$			36.0000i			1.57870i
$521$			12.0000i			0.525730i	0.964833	+	0.262865i	$0.0846673\pi$
$521$			12.0000i			0.525730i	−0.964833	+	0.262865i	$0.915333\pi$
$522$		−	6.00000i		−	0.262613i
$523$	−4.00000			−0.174908			−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000			−0.174908			−0.0874539	+	0.996169i	$0.527873\pi$
$524$			6.00000i			0.262111i
$525$	4.00000			0.174574
$526$	8.00000			0.348817
$527$	16.0000	−	4.00000i	0.696971	−	0.174243i
$528$		0			0
$529$	−41.0000			−1.78261
$530$			12.0000i			0.521247i
$531$	12.0000			0.520756
$532$			16.0000i			0.693688i
$533$			12.0000i			0.519778i
$534$		−	20.0000i		−	0.865485i
$535$	8.00000			0.345870
$536$	24.0000			1.03664
$537$		−	24.0000i		−	1.03568i
$538$			32.0000i			1.37962i
$539$		0			0
$540$	−8.00000			−0.344265
$541$		−	20.0000i		−	0.859867i	−0.902861	−	0.429934i	$-0.858537\pi$
$541$		−	20.0000i		−	0.859867i	0.902861	−	0.429934i	$-0.141463\pi$
$542$		0			0
$543$	4.00000			0.171656
$544$	5.00000	+	20.0000i	0.214373	+	0.857493i
$545$	20.0000			0.856706
$546$	−24.0000			−1.02711
$547$			44.0000i			1.88130i	0.339372	+	0.940652i	$0.389785\pi$
$547$			44.0000i			1.88130i	−0.339372	+	0.940652i	$0.610215\pi$
$548$	18.0000			0.768922
$549$			8.00000i			0.341432i
$550$		0			0
$551$		−	48.0000i		−	2.04487i
$552$	−48.0000			−2.04302
$553$	−28.0000			−1.19068
$554$			28.0000i			1.18961i
$555$		0			0
$556$			8.00000i			0.339276i
$557$	10.0000			0.423714			0.211857	−	0.977301i	$-0.432049\pi$
$557$	10.0000			0.423714			0.211857	+	0.977301i	$0.432049\pi$
$558$			4.00000i			0.169334i
$559$		0			0
$560$	4.00000			0.169031
$561$		0			0
$562$	10.0000			0.421825
$563$	36.0000			1.51722			0.758610	−	0.651546i	$-0.225879\pi$
$563$	36.0000			1.51722			0.758610	+	0.651546i	$0.225879\pi$
$564$		−	2.00000i		−	0.0842152i
$565$	28.0000			1.17797
$566$		−	26.0000i		−	1.09286i
$567$		−	22.0000i		−	0.923913i
$568$		−	30.0000i		−	1.25877i
$569$	−14.0000			−0.586911			−0.293455	−	0.955973i	$-0.594805\pi$
$569$	−14.0000			−0.586911			−0.293455	+	0.955973i	$0.594805\pi$
$570$	−32.0000			−1.34033
$571$			26.0000i			1.08807i	0.839064	+	0.544033i	$0.183103\pi$
$571$			26.0000i			1.08807i	−0.839064	+	0.544033i	$0.816897\pi$
$572$		0			0
$573$		−	16.0000i		−	0.668410i
$574$	4.00000			0.166957
$575$			8.00000i			0.333623i
$576$	−7.00000			−0.291667
$577$	−34.0000			−1.41544			−0.707719	−	0.706494i	$-0.750276\pi$
$577$	−34.0000			−1.41544			−0.707719	+	0.706494i	$0.750276\pi$
$578$	−15.0000	+	8.00000i	−0.623918	+	0.332756i
$579$	4.00000			0.166234
$580$	12.0000			0.498273
$581$		−	8.00000i		−	0.331896i
$582$	16.0000			0.663221
$583$		0			0
$584$			6.00000i			0.248282i
$585$			12.0000i			0.496139i
$586$	30.0000			1.23929
$587$	−32.0000			−1.32078			−0.660391	−	0.750922i	$-0.729609\pi$
$587$	−32.0000			−1.32078			−0.660391	+	0.750922i	$0.729609\pi$
$588$			6.00000i			0.247436i
$589$			32.0000i			1.31854i
$590$		−	24.0000i		−	0.988064i
$591$	24.0000			0.987228
$592$		0			0
$593$	34.0000			1.39621			0.698106	−	0.715994i	$-0.254026\pi$
$593$	34.0000			1.39621			0.698106	+	0.715994i	$0.254026\pi$
$594$		0			0
$595$	−4.00000	−	16.0000i	−0.163984	−	0.655936i
$596$	−18.0000			−0.737309
$597$	32.0000			1.30967
$598$		−	48.0000i		−	1.96287i
$599$	16.0000			0.653742			0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000			0.653742			0.326871	+	0.945069i	$0.394006\pi$
$600$			6.00000i			0.244949i
$601$		−	16.0000i		−	0.652654i	−0.945257	−	0.326327i	$-0.894189\pi$
$601$		−	16.0000i		−	0.652654i	0.945257	−	0.326327i	$-0.105811\pi$
$602$		0			0
$603$	8.00000			0.325785
$604$	16.0000			0.651031
$605$			22.0000i			0.894427i
$606$			12.0000i			0.487467i
$607$			36.0000i			1.46119i	0.682808	+	0.730597i	$0.260758\pi$
$607$			36.0000i			1.46119i	−0.682808	+	0.730597i	$0.739242\pi$
$608$	−40.0000			−1.62221
$609$			24.0000i			0.972529i
$610$	16.0000			0.647821
$611$	6.00000			0.242734
$612$	1.00000	+	4.00000i	0.0404226	+	0.161690i
$613$	6.00000			0.242338			0.121169	−	0.992632i	$-0.461336\pi$
$613$	6.00000			0.242338			0.121169	+	0.992632i	$0.461336\pi$
$614$	28.0000			1.12999
$615$		−	8.00000i		−	0.322591i
$616$		0			0
$617$		−	8.00000i		−	0.322068i	−0.986949	−	0.161034i	$-0.948517\pi$
$617$		−	8.00000i		−	0.322068i	0.986949	−	0.161034i	$-0.0514829\pi$
$618$		−	16.0000i		−	0.643614i
$619$			6.00000i			0.241160i	0.992704	+	0.120580i	$0.0384755\pi$
$619$			6.00000i			0.241160i	−0.992704	+	0.120580i	$0.961525\pi$
$620$	−8.00000			−0.321288
$621$	32.0000			1.28412
$622$		−	16.0000i		−	0.641542i
$623$			20.0000i			0.801283i
$624$		−	12.0000i		−	0.480384i
$625$	−19.0000			−0.760000
$626$			14.0000i			0.559553i
$627$		0			0
$628$	14.0000			0.558661
$629$		0			0
$630$	4.00000			0.159364
$631$	−8.00000			−0.318475			−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000			−0.318475			−0.159237	+	0.987240i	$0.550904\pi$
$632$		−	42.0000i		−	1.67067i
$633$	−8.00000			−0.317971
$634$		−	6.00000i		−	0.238290i
$635$		−	24.0000i		−	0.952411i
$636$			12.0000i			0.475831i
$637$	−18.0000			−0.713186
$638$		0			0
$639$		−	10.0000i		−	0.395594i
$640$		−	6.00000i		−	0.237171i
$641$		−	22.0000i		−	0.868948i	−0.900684	−	0.434474i	$-0.856934\pi$
$641$		−	22.0000i		−	0.868948i	0.900684	−	0.434474i	$-0.143066\pi$
$642$	−8.00000			−0.315735
$643$			14.0000i			0.552106i	0.961142	+	0.276053i	$0.0890266\pi$
$643$			14.0000i			0.552106i	−0.961142	+	0.276053i	$0.910973\pi$
$644$	16.0000			0.630488
$645$		0			0
$646$	−8.00000	−	32.0000i	−0.314756	−	1.25902i
$647$	−24.0000			−0.943537			−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000			−0.943537			−0.471769	+	0.881722i	$0.656384\pi$
$648$	33.0000			1.29636
$649$		0			0
$650$	−6.00000			−0.235339
$651$		−	16.0000i		−	0.627089i
$652$		−	8.00000i		−	0.313304i
$653$			24.0000i			0.939193i	0.882881	+	0.469596i	$0.155601\pi$
$653$			24.0000i			0.939193i	−0.882881	+	0.469596i	$0.844399\pi$
$654$	−20.0000			−0.782062
$655$	12.0000			0.468879
$656$			2.00000i			0.0780869i
$657$			2.00000i			0.0780274i
$658$		−	2.00000i		−	0.0779681i
$659$	−44.0000			−1.71400			−0.856998	−	0.515319i	$-0.827673\pi$
$659$	−44.0000			−1.71400			−0.856998	+	0.515319i	$0.827673\pi$
$660$		0			0
$661$	2.00000			0.0777910			0.0388955	−	0.999243i	$-0.487616\pi$
$661$	2.00000			0.0777910			0.0388955	+	0.999243i	$0.487616\pi$
$662$	4.00000			0.155464
$663$	−48.0000	+	12.0000i	−1.86417	+	0.466041i
$664$	12.0000			0.465690
$665$	32.0000			1.24091
$666$		0			0
$667$	−48.0000			−1.85857
$668$			12.0000i			0.464294i
$669$		−	16.0000i		−	0.618596i
$670$		−	16.0000i		−	0.618134i
$671$		0			0
$672$	20.0000			0.771517
$673$		−	46.0000i		−	1.77317i	−0.462566	−	0.886585i	$-0.653071\pi$
$673$		−	46.0000i		−	1.77317i	0.462566	−	0.886585i	$-0.346929\pi$
$674$			36.0000i			1.38667i
$675$		−	4.00000i		−	0.153960i
$676$	−23.0000			−0.884615
$677$		−	2.00000i		−	0.0768662i	−0.999261	−	0.0384331i	$-0.987763\pi$
$677$		−	2.00000i		−	0.0768662i	0.999261	−	0.0384331i	$-0.0122367\pi$
$678$	−28.0000			−1.07533
$679$	−16.0000			−0.614024
$680$	24.0000	−	6.00000i	0.920358	−	0.230089i
$681$	24.0000			0.919682
$682$		0			0
$683$			6.00000i			0.229584i	0.993390	+	0.114792i	$0.0366201\pi$
$683$			6.00000i			0.229584i	−0.993390	+	0.114792i	$0.963380\pi$
$684$	−8.00000			−0.305888
$685$		−	36.0000i		−	1.37549i
$686$			20.0000i			0.763604i
$687$		−	20.0000i		−	0.763048i
$688$		0			0
$689$	−36.0000			−1.37149
$690$			32.0000i			1.21822i
$691$		−	20.0000i		−	0.760836i	−0.924815	−	0.380418i	$-0.875780\pi$
$691$		−	20.0000i		−	0.760836i	0.924815	−	0.380418i	$-0.124220\pi$
$692$		−	8.00000i		−	0.304114i
$693$		0			0
$694$			22.0000i			0.835109i
$695$	16.0000			0.606915
$696$	−36.0000			−1.36458
$697$	8.00000	−	2.00000i	0.303022	−	0.0757554i
$698$	−18.0000			−0.681310
$699$	36.0000			1.36165
$700$		−	2.00000i		−	0.0755929i
$701$	−30.0000			−1.13308			−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000			−1.13308			−0.566542	+	0.824033i	$0.691719\pi$
$702$			24.0000i			0.905822i
$703$		0			0
$704$		0			0
$705$	−4.00000			−0.150649
$706$	−18.0000			−0.677439
$707$		−	12.0000i		−	0.451306i
$708$		−	24.0000i		−	0.901975i
$709$		−	36.0000i		−	1.35201i	−0.736898	−	0.676004i	$-0.763710\pi$
$709$		−	36.0000i		−	1.35201i	0.736898	−	0.676004i	$-0.236290\pi$
$710$	−20.0000			−0.750587
$711$		−	14.0000i		−	0.525041i
$712$	−30.0000			−1.12430
$713$	32.0000			1.19841
$714$	4.00000	+	16.0000i	0.149696	+	0.598785i
$715$		0			0
$716$	−12.0000			−0.448461
$717$		0			0
$718$	−24.0000			−0.895672
$719$		−	38.0000i		−	1.41716i	−0.705630	−	0.708580i	$-0.749336\pi$
$719$		−	38.0000i		−	1.41716i	0.705630	−	0.708580i	$-0.250664\pi$
$720$			2.00000i			0.0745356i
$721$			16.0000i			0.595871i
$722$	45.0000			1.67473
$723$	24.0000			0.892570
$724$		−	2.00000i		−	0.0743294i
$725$			6.00000i			0.222834i
$726$		−	22.0000i		−	0.816497i
$727$	−28.0000			−1.03846			−0.519231	−	0.854634i	$-0.673782\pi$
$727$	−28.0000			−1.03846			−0.519231	+	0.854634i	$0.673782\pi$
$728$			36.0000i			1.33425i
$729$	−13.0000			−0.481481
$730$	4.00000			0.148047
$731$		0			0
$732$	16.0000			0.591377
$733$	−6.00000			−0.221615			−0.110808	−	0.993842i	$-0.535344\pi$
$733$	−6.00000			−0.221615			−0.110808	+	0.993842i	$0.535344\pi$
$734$		−	12.0000i		−	0.442928i
$735$	12.0000			0.442627
$736$			40.0000i			1.47442i
$737$		0			0
$738$			2.00000i			0.0736210i
$739$	−36.0000			−1.32428			−0.662141	−	0.749380i	$-0.730352\pi$
$739$	−36.0000			−1.32428			−0.662141	+	0.749380i	$0.730352\pi$
$740$		0			0
$741$		−	96.0000i		−	3.52665i
$742$			12.0000i			0.440534i
$743$			8.00000i			0.293492i	0.989174	+	0.146746i	$0.0468799\pi$
$743$			8.00000i			0.293492i	−0.989174	+	0.146746i	$0.953120\pi$
$744$	24.0000			0.879883
$745$			36.0000i			1.31894i
$746$	−14.0000			−0.512576
$747$	4.00000			0.146352
$748$		0			0
$749$	8.00000			0.292314
$750$	24.0000			0.876356
$751$		−	32.0000i		−	1.16770i	−0.811863	−	0.583848i	$-0.801546\pi$
$751$		−	32.0000i		−	1.16770i	0.811863	−	0.583848i	$-0.198454\pi$
$752$	1.00000			0.0364662
$753$		−	8.00000i		−	0.291536i
$754$		−	36.0000i		−	1.31104i
$755$		−	32.0000i		−	1.16460i
$756$	−8.00000			−0.290957
$757$	2.00000			0.0726912			0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000			0.0726912			0.0363456	+	0.999339i	$0.488428\pi$
$758$			10.0000i			0.363216i
$759$		0			0
$760$			48.0000i			1.74114i
$761$	−42.0000			−1.52250			−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000			−1.52250			−0.761249	+	0.648459i	$0.775414\pi$
$762$			24.0000i			0.869428i
$763$	20.0000			0.724049
$764$	−8.00000			−0.289430
$765$	8.00000	−	2.00000i	0.289241	−	0.0723102i
$766$	16.0000			0.578103
$767$	72.0000			2.59977
$768$			34.0000i			1.22687i
$769$	−14.0000			−0.504853			−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000			−0.504853			−0.252426	+	0.967616i	$0.581229\pi$
$770$		0			0
$771$		−	44.0000i		−	1.58462i
$772$		−	2.00000i		−	0.0719816i
$773$	26.0000			0.935155			0.467578	−	0.883952i	$-0.345127\pi$
$773$	26.0000			0.935155			0.467578	+	0.883952i	$0.345127\pi$
$774$		0			0
$775$		−	4.00000i		−	0.143684i
$776$		−	24.0000i		−	0.861550i
$777$		0			0
$778$	18.0000			0.645331
$779$			16.0000i			0.573259i
$780$	24.0000			0.859338
$781$		0			0
$782$	−32.0000	+	8.00000i	−1.14432	+	0.286079i
$783$	24.0000			0.857690
$784$	−3.00000			−0.107143
$785$		−	28.0000i		−	0.999363i
$786$	−12.0000			−0.428026
$787$			20.0000i			0.712923i	0.934310	+	0.356462i	$0.116017\pi$
$787$			20.0000i			0.712923i	−0.934310	+	0.356462i	$0.883983\pi$
$788$		−	12.0000i		−	0.427482i
$789$		−	16.0000i		−	0.569615i
$790$	−28.0000			−0.996195
$791$	28.0000			0.995565
$792$		0			0
$793$			48.0000i			1.70453i
$794$			32.0000i			1.13564i
$795$	24.0000			0.851192
$796$		−	16.0000i		−	0.567105i
$797$	30.0000			1.06265			0.531327	−	0.847167i	$-0.321693\pi$
$797$	30.0000			1.06265			0.531327	+	0.847167i	$0.321693\pi$
$798$	−32.0000			−1.13279
$799$	−1.00000	−	4.00000i	−0.0353775	−	0.141510i
$800$	5.00000			0.176777
$801$	−10.0000			−0.353333
$802$		−	28.0000i		−	0.988714i
$803$		0			0
$804$		−	16.0000i		−	0.564276i
$805$		−	32.0000i		−	1.12785i
$806$			24.0000i			0.845364i
$807$	64.0000			2.25291
$808$	18.0000			0.633238
$809$			26.0000i			0.914111i	0.889438	+	0.457056i	$0.151096\pi$
$809$			26.0000i			0.914111i	−0.889438	+	0.457056i	$0.848904\pi$
$810$		−	22.0000i		−	0.773001i
$811$		−	54.0000i		−	1.89620i	−0.317978	−	0.948098i	$-0.603004\pi$
$811$		−	54.0000i		−	1.89620i	0.317978	−	0.948098i	$-0.396996\pi$
$812$	12.0000			0.421117
$813$		0			0
$814$		0			0
$815$	−16.0000			−0.560456
$816$	−8.00000	+	2.00000i	−0.280056	+	0.0700140i
$817$		0			0
$818$	−38.0000			−1.32864
$819$			12.0000i			0.419314i
$820$	−4.00000			−0.139686
$821$		−	42.0000i		−	1.46581i	−0.680331	−	0.732905i	$-0.738164\pi$
$821$		−	42.0000i		−	1.46581i	0.680331	−	0.732905i	$-0.261836\pi$
$822$			36.0000i			1.25564i
$823$		−	50.0000i		−	1.74289i	−0.490493	−	0.871445i	$-0.663183\pi$
$823$		−	50.0000i		−	1.74289i	0.490493	−	0.871445i	$-0.336817\pi$
$824$	−24.0000			−0.836080
$825$		0			0
$826$		−	24.0000i		−	0.835067i
$827$		−	2.00000i		−	0.0695468i	−0.999395	−	0.0347734i	$-0.988929\pi$
$827$		−	2.00000i		−	0.0695468i	0.999395	−	0.0347734i	$-0.0110710\pi$
$828$			8.00000i			0.278019i
$829$	−2.00000			−0.0694629			−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000			−0.0694629			−0.0347314	+	0.999397i	$0.511058\pi$
$830$		−	8.00000i		−	0.277684i
$831$	56.0000			1.94262
$832$	−42.0000			−1.45609
$833$	3.00000	+	12.0000i	0.103944	+	0.415775i
$834$	−16.0000			−0.554035
$835$	24.0000			0.830554
$836$		0			0
$837$	−16.0000			−0.553041
$838$		−	12.0000i		−	0.414533i
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		−	24.0000i		−	0.828079i
$841$	−7.00000			−0.241379
$842$	−6.00000			−0.206774
$843$		−	20.0000i		−	0.688837i
$844$			4.00000i			0.137686i
$845$			46.0000i			1.58245i
$846$	1.00000			0.0343807
$847$			22.0000i			0.755929i
$848$	−6.00000			−0.206041
$849$	−52.0000			−1.78464
$850$	1.00000	+	4.00000i	0.0342997	+	0.137199i
$851$		0			0
$852$	−20.0000			−0.685189
$853$			16.0000i			0.547830i	0.961754	+	0.273915i	$0.0883186\pi$
$853$			16.0000i			0.547830i	−0.961754	+	0.273915i	$0.911681\pi$
$854$	16.0000			0.547509
$855$			16.0000i			0.547188i
$856$			12.0000i			0.410152i
$857$			18.0000i			0.614868i	0.951569	+	0.307434i	$0.0994704\pi$
$857$			18.0000i			0.614868i	−0.951569	+	0.307434i	$0.900530\pi$
$858$		0			0
$859$	44.0000			1.50126			0.750630	−	0.660722i	$-0.229750\pi$
$859$	44.0000			1.50126			0.750630	+	0.660722i	$0.229750\pi$
$860$		0			0
$861$		−	8.00000i		−	0.272639i
$862$		−	18.0000i		−	0.613082i
$863$	56.0000			1.90626			0.953131	−	0.302558i	$-0.0978405\pi$
$863$	56.0000			1.90626			0.953131	+	0.302558i	$0.0978405\pi$
$864$		−	20.0000i		−	0.680414i
$865$	−16.0000			−0.544016
$866$	14.0000			0.475739
$867$	16.0000	+	30.0000i	0.543388	+	1.01885i
$868$	−8.00000			−0.271538
$869$		0			0
$870$			24.0000i			0.813676i
$871$	48.0000			1.62642
$872$			30.0000i			1.01593i
$873$		−	8.00000i		−	0.270759i
$874$		−	64.0000i		−	2.16483i
$875$	−24.0000			−0.811348
$876$	4.00000			0.135147
$877$			46.0000i			1.55331i	0.629926	+	0.776655i	$0.283085\pi$
$877$			46.0000i			1.55331i	−0.629926	+	0.776655i	$0.716915\pi$
$878$			6.00000i			0.202490i
$879$		−	60.0000i		−	2.02375i
$880$		0			0
$881$		−	30.0000i		−	1.01073i	−0.862907	−	0.505363i	$-0.831359\pi$
$881$		−	30.0000i		−	1.01073i	0.862907	−	0.505363i	$-0.168641\pi$
$882$	−3.00000			−0.101015
$883$	20.0000			0.673054			0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000			0.673054			0.336527	+	0.941674i	$0.390748\pi$
$884$	6.00000	+	24.0000i	0.201802	+	0.807207i
$885$	−48.0000			−1.61350
$886$	−24.0000			−0.806296
$887$		−	32.0000i		−	1.07445i	−0.843437	−	0.537227i	$-0.819472\pi$
$887$		−	32.0000i		−	1.07445i	0.843437	−	0.537227i	$-0.180528\pi$
$888$		0			0
$889$		−	24.0000i		−	0.804934i
$890$			20.0000i			0.670402i
$891$		0			0
$892$	−8.00000			−0.267860
$893$	8.00000			0.267710
$894$		−	36.0000i		−	1.20402i
$895$			24.0000i			0.802232i
$896$		−	6.00000i		−	0.200446i
$897$	−96.0000			−3.20535
$898$		−	18.0000i		−	0.600668i
$899$	24.0000			0.800445
$900$	1.00000			0.0333333
$901$	6.00000	+	24.0000i	0.199889	+	0.799556i
$902$		0			0
$903$		0			0
$904$			42.0000i			1.39690i
$905$	−4.00000			−0.132964
$906$			32.0000i			1.06313i
$907$			18.0000i			0.597680i	0.954303	+	0.298840i	$0.0965997\pi$
$907$			18.0000i			0.597680i	−0.954303	+	0.298840i	$0.903400\pi$
$908$		−	12.0000i		−	0.398234i
$909$	6.00000			0.199007
$910$	24.0000			0.795592
$911$		−	18.0000i		−	0.596367i	−0.954509	−	0.298183i	$-0.903619\pi$
$911$		−	18.0000i		−	0.596367i	0.954509	−	0.298183i	$-0.0963807\pi$
$912$		−	16.0000i		−	0.529813i
$913$		0			0
$914$	10.0000			0.330771
$915$		−	32.0000i		−	1.05789i
$916$	−10.0000			−0.330409
$917$	12.0000			0.396275
$918$	16.0000	−	4.00000i	0.528079	−	0.132020i
$919$	4.00000			0.131948			0.0659739	−	0.997821i	$-0.478985\pi$
$919$	4.00000			0.131948			0.0659739	+	0.997821i	$0.478985\pi$
$920$	48.0000			1.58251
$921$		−	56.0000i		−	1.84526i
$922$	14.0000			0.461065
$923$		−	60.0000i		−	1.97492i
$924$		0			0
$925$		0			0
$926$	32.0000			1.05159
$927$	−8.00000			−0.262754
$928$			30.0000i			0.984798i
$929$			20.0000i			0.656179i	0.944647	+	0.328089i	$0.106405\pi$
$929$			20.0000i			0.656179i	−0.944647	+	0.328089i	$0.893595\pi$
$930$		−	16.0000i		−	0.524661i
$931$	−24.0000			−0.786568
$932$		−	18.0000i		−	0.589610i
$933$	−32.0000			−1.04763
$934$	4.00000			0.130884
$935$		0			0
$936$	−18.0000			−0.588348
$937$	34.0000			1.11073			0.555366	−	0.831606i	$-0.312578\pi$
$937$	34.0000			1.11073			0.555366	+	0.831606i	$0.312578\pi$
$938$		−	16.0000i		−	0.522419i
$939$	28.0000			0.913745
$940$			2.00000i			0.0652328i
$941$			8.00000i			0.260793i	0.991462	+	0.130396i	$0.0416250\pi$
$941$			8.00000i			0.260793i	−0.991462	+	0.130396i	$0.958375\pi$
$942$			28.0000i			0.912289i
$943$	16.0000			0.521032
$944$	12.0000			0.390567
$945$			16.0000i			0.520480i
$946$		0			0
$947$			42.0000i			1.36482i	0.730971	+	0.682408i	$0.239067\pi$
$947$			42.0000i			1.36482i	−0.730971	+	0.682408i	$0.760933\pi$
$948$	−28.0000			−0.909398
$949$			12.0000i			0.389536i
$950$	−8.00000			−0.259554
$951$	−12.0000			−0.389127
$952$	24.0000	−	6.00000i	0.777844	−	0.194461i
$953$	−34.0000			−1.10137			−0.550684	−	0.834714i	$-0.685633\pi$
$953$	−34.0000			−1.10137			−0.550684	+	0.834714i	$0.685633\pi$
$954$	−6.00000			−0.194257
$955$			16.0000i			0.517748i
$956$		0			0
$957$		0			0
$958$			6.00000i			0.193851i
$959$		−	36.0000i		−	1.16250i
$960$	28.0000			0.903696
$961$	15.0000			0.483871
$962$		0			0
$963$			4.00000i			0.128898i
$964$		−	12.0000i		−	0.386494i
$965$	−4.00000			−0.128765
$966$			32.0000i			1.02958i
$967$	16.0000			0.514525			0.257263	−	0.966342i	$-0.417179\pi$
$967$	16.0000			0.514525			0.257263	+	0.966342i	$0.417179\pi$
$968$	−33.0000			−1.06066
$969$	−64.0000	+	16.0000i	−2.05598	+	0.513994i
$970$	−16.0000			−0.513729
$971$	−24.0000			−0.770197			−0.385098	−	0.922876i	$-0.625832\pi$
$971$	−24.0000			−0.770197			−0.385098	+	0.922876i	$0.625832\pi$
$972$		−	10.0000i		−	0.320750i
$973$	16.0000			0.512936
$974$		−	26.0000i		−	0.833094i
$975$			12.0000i			0.384308i
$976$			8.00000i			0.256074i
$977$	38.0000			1.21573			0.607864	−	0.794041i	$-0.292027\pi$
$977$	38.0000			1.21573			0.607864	+	0.794041i	$0.292027\pi$
$978$	16.0000			0.511624
$979$		0			0
$980$		−	6.00000i		−	0.191663i
$981$			10.0000i			0.319275i
$982$	−36.0000			−1.14881
$983$			24.0000i			0.765481i	0.923856	+	0.382741i	$0.125020\pi$
$983$			24.0000i			0.765481i	−0.923856	+	0.382741i	$0.874980\pi$
$984$	12.0000			0.382546
$985$	−24.0000			−0.764704
$986$	−24.0000	+	6.00000i	−0.764316	+	0.191079i
$987$	−4.00000			−0.127321
$988$	−48.0000			−1.52708
$989$		0			0
$990$		0			0
$991$			18.0000i			0.571789i	0.958261	+	0.285894i	$0.0922907\pi$
$991$			18.0000i			0.571789i	−0.958261	+	0.285894i	$0.907709\pi$
$992$		−	20.0000i		−	0.635001i
$993$		−	8.00000i		−	0.253872i
$994$	−20.0000			−0.634361
$995$	−32.0000			−1.01447
$996$		−	8.00000i		−	0.253490i
$997$			2.00000i			0.0633406i	0.999498	+	0.0316703i	$0.0100827\pi$
$997$			2.00000i			0.0633406i	−0.999498	+	0.0316703i	$0.989917\pi$
$998$			16.0000i			0.506471i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	799.2.b.a.424.1	✓	2
17.16	even	2	inner	799.2.b.a.424.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
799.2.b.a.424.1	✓	2	1.1	even	1	trivial
799.2.b.a.424.2	yes	2	17.16	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 799 = 17 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	799.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.00000i q^{3} -1.00000 q^{4} +2.00000i q^{5} -2.00000i q^{6} +2.00000i q^{7} -3.00000 q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.00000i q^{3} -1.00000 q^{4} +2.00000i q^{5} -2.00000i q^{6} +2.00000i q^{7} -3.00000 q^{8} -1.00000 q^{9} +2.00000i q^{10} +2.00000i q^{12} -6.00000 q^{13} +2.00000i q^{14} +4.00000 q^{15} -1.00000 q^{16} +(1.00000 + 4.00000i) q^{17} -1.00000 q^{18} -8.00000 q^{19} -2.00000i q^{20} +4.00000 q^{21} +8.00000i q^{23} +6.00000i q^{24} +1.00000 q^{25} -6.00000 q^{26} -4.00000i q^{27} -2.00000i q^{28} +6.00000i q^{29} +4.00000 q^{30} -4.00000i q^{31} +5.00000 q^{32} +(1.00000 + 4.00000i) q^{34} -4.00000 q^{35} +1.00000 q^{36} -8.00000 q^{38} +12.0000i q^{39} -6.00000i q^{40} -2.00000i q^{41} +4.00000 q^{42} -2.00000i q^{45} +8.00000i q^{46} -1.00000 q^{47} +2.00000i q^{48} +3.00000 q^{49} +1.00000 q^{50} +(8.00000 - 2.00000i) q^{51} +6.00000 q^{52} +6.00000 q^{53} -4.00000i q^{54} -6.00000i q^{56} +16.0000i q^{57} +6.00000i q^{58} -12.0000 q^{59} -4.00000 q^{60} -8.00000i q^{61} -4.00000i q^{62} -2.00000i q^{63} +7.00000 q^{64} -12.0000i q^{65} -8.00000 q^{67} +(-1.00000 - 4.00000i) q^{68} +16.0000 q^{69} -4.00000 q^{70} +10.0000i q^{71} +3.00000 q^{72} -2.00000i q^{73} -2.00000i q^{75} +8.00000 q^{76} +12.0000i q^{78} +14.0000i q^{79} -2.00000i q^{80} -11.0000 q^{81} -2.00000i q^{82} -4.00000 q^{83} -4.00000 q^{84} +(-8.00000 + 2.00000i) q^{85} +12.0000 q^{87} +10.0000 q^{89} -2.00000i q^{90} -12.0000i q^{91} -8.00000i q^{92} -8.00000 q^{93} -1.00000 q^{94} -16.0000i q^{95} -10.0000i q^{96} +8.00000i q^{97} +3.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{4} - 6 q^{8} - 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 - 2 * q^4 - 6 * q^8 - 2 * q^9	\( 2 q + 2 q^{2} - 2 q^{4} - 6 q^{8} - 2 q^{9} - 12 q^{13} + 8 q^{15} - 2 q^{16} + 2 q^{17} - 2 q^{18} - 16 q^{19} + 8 q^{21} + 2 q^{25} - 12 q^{26} + 8 q^{30} + 10 q^{32} + 2 q^{34} - 8 q^{35} + 2 q^{36} - 16 q^{38} + 8 q^{42} - 2 q^{47} + 6 q^{49} + 2 q^{50} + 16 q^{51} + 12 q^{52} + 12 q^{53} - 24 q^{59} - 8 q^{60} + 14 q^{64} - 16 q^{67} - 2 q^{68} + 32 q^{69} - 8 q^{70} + 6 q^{72} + 16 q^{76} - 22 q^{81} - 8 q^{83} - 8 q^{84} - 16 q^{85} + 24 q^{87} + 20 q^{89} - 16 q^{93} - 2 q^{94} + 6 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^4 - 6 * q^8 - 2 * q^9 - 12 * q^13 + 8 * q^15 - 2 * q^16 + 2 * q^17 - 2 * q^18 - 16 * q^19 + 8 * q^21 + 2 * q^25 - 12 * q^26 + 8 * q^30 + 10 * q^32 + 2 * q^34 - 8 * q^35 + 2 * q^36 - 16 * q^38 + 8 * q^42 - 2 * q^47 + 6 * q^49 + 2 * q^50 + 16 * q^51 + 12 * q^52 + 12 * q^53 - 24 * q^59 - 8 * q^60 + 14 * q^64 - 16 * q^67 - 2 * q^68 + 32 * q^69 - 8 * q^70 + 6 * q^72 + 16 * q^76 - 22 * q^81 - 8 * q^83 - 8 * q^84 - 16 * q^85 + 24 * q^87 + 20 * q^89 - 16 * q^93 - 2 * q^94 + 6 * q^98

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